1,002 research outputs found
Recent Results on Near-Best Spline Quasi-Interpolants
Roughly speaking, a near-best (abbr. NB) quasi-interpolant (abbr. QI) is an
approximation operator of the form where the 's are B-splines and the 's
are linear discrete or integral forms acting on the given function . These
forms depend on a finite number of coefficients which are the components of
vectors for . The index refers to this sequence of
vectors. In order that for all polynomials belonging to some
subspace included in the space of splines generated by the 's, each
vector must lie in an affine subspace , i.e. satisfy some
linear constraints. However there remain some degrees of freedom which are used
to minimize for each . It is easy to
prove that is an upper bound of
: thus, instead of minimizing the infinite norm of
, which is a difficult problem, we minimize an upper bound of this norm,
which is much easier to do. Moreover, the latter problem has always at least
one solution, which is associated with a NB QI. In the first part of the paper,
we give a survey on NB univariate or bivariate spline QIs defined on uniform or
non-uniform partitions and already studied by the author and coworkers. In the
second part, we give some new results, mainly on univariate and bivariate
integral QIs on {\sl non-uniform} partitions: in that case, NB QIs are more
difficult to characterize and the optimal properties strongly depend on the
geometry of the partition. Therefore we have restricted our study to QIs having
interesting shape properties and/or infinite norms uniformly bounded
independently of the partition
Multipatch Approximation of the de Rham Sequence and its Traces in Isogeometric Analysis
We define a conforming B-spline discretisation of the de Rham complex on
multipatch geometries. We introduce and analyse the properties of interpolation
operators onto these spaces which commute w.r.t. the surface differential
operators. Using these results as a basis, we derive new convergence results of
optimal order w.r.t. the respective energy spaces and provide approximation
properties of the spline discretisations of trace spaces for application in the
theory of isogeometric boundary element methods. Our analysis allows for a
straightforward generalisation to finite element methods
Piecewise affine approximations for functions of bounded variation
BV functions cannot be approximated well by piecewise constant functions, but
this work will show that a good approximation is still possible with
(countably) piecewise affine functions. In particular, this approximation is
area-strictly close to the original function and the -difference
between the traces of the original and approximating functions on a substantial
part of the mesh can be made arbitrarily small. Necessarily, the mesh needs to
be adapted to the singularities of the BV function to be approximated, and
consequently, the proof is based on a blow-up argument together with explicit
constructions of the mesh. In the case of -Sobolev functions
we establish an optimal -error estimate for approximation by
piecewise affine functions on uniform regular triangulations. The piecewise
affine functions are standard quasi-interpolants obtained by mollification and
Lagrange interpolation on the nodes of triangulations, and the main new
contribution here compared to for instance Cl\'{e}ment (RAIRO Analyse
Num\'{e}rique 9 (1975), no.~R-2, 77--84) and Verf\"{u}rth (M2AN
Math.~Model.~Numer.~Anal. 33 (1999), no. 4, 695-713) is that our error
estimates are in the -norm rather than merely the
-norm.Comment: 14 pages, 1 figur
Multilevel Sparse Grid Methods for Elliptic Partial Differential Equations with Random Coefficients
Stochastic sampling methods are arguably the most direct and least intrusive
means of incorporating parametric uncertainty into numerical simulations of
partial differential equations with random inputs. However, to achieve an
overall error that is within a desired tolerance, a large number of sample
simulations may be required (to control the sampling error), each of which may
need to be run at high levels of spatial fidelity (to control the spatial
error). Multilevel sampling methods aim to achieve the same accuracy as
traditional sampling methods, but at a reduced computational cost, through the
use of a hierarchy of spatial discretization models. Multilevel algorithms
coordinate the number of samples needed at each discretization level by
minimizing the computational cost, subject to a given error tolerance. They can
be applied to a variety of sampling schemes, exploit nesting when available,
can be implemented in parallel and can be used to inform adaptive spatial
refinement strategies. We extend the multilevel sampling algorithm to sparse
grid stochastic collocation methods, discuss its numerical implementation and
demonstrate its efficiency both theoretically and by means of numerical
examples
A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations
A new projection method based on radial basis functions (RBFs) is presented
for discretizing the incompressible unsteady Stokes equations in irregular
geometries. The novelty of the method comes from the application of a new
technique for computing the Leray-Helmholtz projection of a vector field using
generalized interpolation with divergence-free and curl-free RBFs. Unlike
traditional projection methods, this new method enables matching both
tangential and normal components of divergence-free vector fields on the domain
boundary. This allows incompressibility of the velocity field to be enforced
without any time-splitting or pressure boundary conditions. Spatial derivatives
are approximated using collocation with global RBFs so that the method only
requires samples of the field at (possibly scattered) nodes over the domain.
Numerical results are presented demonstrating high-order convergence in both
space (between 5th and 6th order) and time (up to 4th order) for some model
problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure
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